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Properties

Label 35.10.f.a
Level $35$
Weight $10$
Character orbit 35.f
Analytic conductor $18.026$
Analytic rank $0$
Dimension $68$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,10,Mod(13,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.13");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.0262542657\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} - 5946 q^{7} + 3472 q^{8} - 83032 q^{11} + 318860 q^{15} - 4744440 q^{16} + 357128 q^{18} + 999732 q^{21} - 4174596 q^{22} - 5819092 q^{23} - 38300 q^{25} + 17370560 q^{28} + 20856500 q^{30}+ \cdots + 8347561548 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −30.0796 + 30.0796i −165.104 + 165.104i − 1297.56i −1175.60 + 755.707i − 9932.50i −6017.78 + 2034.69i 23629.3 + 23629.3i − 34835.5i 12630.2 − 58092.8i
13.2 −30.0796 + 30.0796i 165.104 − 165.104i − 1297.56i 1175.60 − 755.707i 9932.50i −2034.69 + 6017.78i 23629.3 + 23629.3i − 34835.5i −12630.2 + 58092.8i
13.3 −28.4613 + 28.4613i −65.3602 + 65.3602i − 1108.09i 920.000 − 1052.01i − 3720.47i 334.479 − 6343.64i 16965.6 + 16965.6i 11139.1i 3757.16 + 56126.0i
13.4 −28.4613 + 28.4613i 65.3602 − 65.3602i − 1108.09i −920.000 + 1052.01i 3720.47i 6343.64 − 334.479i 16965.6 + 16965.6i 11139.1i −3757.16 − 56126.0i
13.5 −22.4049 + 22.4049i −64.8175 + 64.8175i − 491.961i 1234.64 + 654.815i − 2904.46i −2084.54 + 6000.69i −448.973 − 448.973i 11280.4i −42333.1 + 12991.0i
13.6 −22.4049 + 22.4049i 64.8175 − 64.8175i − 491.961i −1234.64 − 654.815i 2904.46i −6000.69 + 2084.54i −448.973 − 448.973i 11280.4i 42333.1 − 12991.0i
13.7 −21.8088 + 21.8088i −80.0540 + 80.0540i − 439.244i −536.173 − 1290.60i − 3491.76i 5525.04 + 3134.88i −1586.71 − 1586.71i 6865.73i 39839.6 + 16453.1i
13.8 −21.8088 + 21.8088i 80.0540 − 80.0540i − 439.244i 536.173 + 1290.60i 3491.76i −3134.88 − 5525.04i −1586.71 − 1586.71i 6865.73i −39839.6 − 16453.1i
13.9 −16.7332 + 16.7332i −158.487 + 158.487i − 47.9996i 572.153 + 1275.06i − 5303.99i 4880.13 − 4066.68i −7764.21 − 7764.21i − 30553.3i −30909.7 − 11761.8i
13.10 −16.7332 + 16.7332i 158.487 − 158.487i − 47.9996i −572.153 − 1275.06i 5303.99i 4066.68 − 4880.13i −7764.21 − 7764.21i − 30553.3i 30909.7 + 11761.8i
13.11 −10.7176 + 10.7176i −94.9252 + 94.9252i 282.266i −1392.70 + 116.269i − 2034.74i −2241.22 − 5943.95i −8512.63 − 8512.63i 1661.42i 13680.3 − 16172.5i
13.12 −10.7176 + 10.7176i 94.9252 − 94.9252i 282.266i 1392.70 − 116.269i 2034.74i 5943.95 + 2241.22i −8512.63 − 8512.63i 1661.42i −13680.3 + 16172.5i
13.13 −8.85794 + 8.85794i −185.026 + 185.026i 355.074i 321.387 − 1360.09i − 3277.89i −4056.41 + 4888.68i −7680.49 − 7680.49i − 48786.0i 9200.74 + 14894.4i
13.14 −8.85794 + 8.85794i 185.026 − 185.026i 355.074i −321.387 + 1360.09i 3277.89i −4888.68 + 4056.41i −7680.49 − 7680.49i − 48786.0i −9200.74 − 14894.4i
13.15 −7.91668 + 7.91668i −12.0397 + 12.0397i 386.652i −979.158 + 997.183i − 190.628i 3026.19 + 5585.32i −7114.34 − 7114.34i 19393.1i −142.700 − 15646.1i
13.16 −7.91668 + 7.91668i 12.0397 − 12.0397i 386.652i 979.158 − 997.183i 190.628i −5585.32 − 3026.19i −7114.34 − 7114.34i 19393.1i 142.700 + 15646.1i
13.17 3.98832 − 3.98832i −120.818 + 120.818i 480.187i 1318.93 − 462.121i 963.720i 5668.35 − 2867.66i 3957.16 + 3957.16i − 9510.88i 3417.22 − 7103.39i
13.18 3.98832 − 3.98832i 120.818 − 120.818i 480.187i −1318.93 + 462.121i − 963.720i 2867.66 − 5668.35i 3957.16 + 3957.16i − 9510.88i −3417.22 + 7103.39i
13.19 4.04483 − 4.04483i −81.7433 + 81.7433i 479.279i 613.563 + 1255.65i 661.275i −6068.65 + 1877.53i 4009.55 + 4009.55i 6319.07i 7560.66 + 2597.14i
13.20 4.04483 − 4.04483i 81.7433 − 81.7433i 479.279i −613.563 − 1255.65i − 661.275i −1877.53 + 6068.65i 4009.55 + 4009.55i 6319.07i −7560.66 − 2597.14i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.10.f.a ✓ 68
5.c odd 4 1 inner 35.10.f.a ✓ 68
7.b odd 2 1 inner 35.10.f.a ✓ 68
35.f even 4 1 inner 35.10.f.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.f.a ✓ 68 1.a even 1 1 trivial
35.10.f.a ✓ 68 5.c odd 4 1 inner
35.10.f.a ✓ 68 7.b odd 2 1 inner
35.10.f.a ✓ 68 35.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(35, [\chi])\).